Transcript Simultaneous_Equations

Simultaneous Equations
•Elimination Method
•Substitution method
•Graphical Method
•Matrix Method
What are they?
• Simply 2 equations
– With 2 unknowns
– Usually x and y
• To SOLVE the equations means we find values of
x and y that
– Satisfy BOTH equations [work in]
– At same time [simultaneously]
Elimination Method
A
B
We have the same
number of y’s in each
2x – y = 1
If
we ADD the equations, the y’s disappear
+
3x + y = 9
5x
= 10
=2
x
2x2–y=1
4–y=1
y=3
Divide both sides by 5
Substitute x = 2 in equation A
Answer
x = 2, y = 3
Elimination Method
A
B
5x + y = 17
3x + y = 11
2x
=6
=3
x
5 x 3 + y = 17
15 + y = 17
y=2
We have the same
number of y’s in each
If we SUBTRACT the equations,
the y’s disappear
Divide both sides by 2
Substitute x = 3 in equation A
Answer
x = 3, y = 2
Elimination Method
A
B
2x + 3y = 9
2x + y = 7
2y = 2
y =1
2x + 3 = 9
2x = 6
x=3
We have the same
number of x’s in each
If we SUBTRACT the equations,
the x’s disappear
Divide both sides by 2
Substitute y = 1 in equation A
Answer
x = 3, y = 1
Elimination Method
A
B
4x - 3y = 14
+
2x + 3y = 16
6x
= 30
x =5
20 – 3y = 14
3y = 6
y=2
We have the same
number of y’s in each
If we ADD the equations,
the y’s disappear
Divide both sides by 6
Substitute x = 5 in equation A
Answer
x = 5, y = 2
Basic steps
• Look at equations
• Same number of x’s or y’s?
• If the sign is different, ADD the equations
otherwise subtract tem
• Then have ONE equation
• Solve this
• Substitute answer to get the other
• CHECK by substitution of BOTH answers
What if NOT same number of x’s or y’s?
A
B
A
3x + y = 10
5x + 2y = 17
If we multiply A by 2 we
get 2y in each
-
6x + 2y = 20
B 5x + 2y = 17
x
=3
In B 5 x 3 + 2y = 17
15 + 2y = 17
y=1
Answer
x = 3, y = 1
What if NOT same number of x’s or y’s?
A
B
4x - 2y = 8
3x + 6y = 21
If we multiply A by 3 we
get 6y in each
A 12x - 6y = 24
B 3x + 6y = 21
15x
= 45
x=3
In B 3 x 3 + 6y = 21
6y = 12
y=2
+
Answer
x = 3, y = 2
…if multiplying 1 equation doesn’t help?
A 3x + 7y = 26
B 5x + 2y = 24
A 15x + 35y = 130
B 15x + 6y = 72
29y = 58
y=2
In B 5x + 2 x 2 = 24
5x = 20
x=4
Multiply A by 5 & B by 3,
we get 15x in each
-
Could multiply A by 2 & B
by 7 to get 14y in each
Answer
x = 4, y = 2
…if multiplying 1 equation doesn’t help?
A 3x - 2y = 7
B 5x + 3y = 37
A 9x – 6y = 21
B 10x + 6y = 74
= 95
19x
x=5
In B 5 x 5 + 3y = 37
3y = 12
y=4
Multiply A by 3 & B by 2,
we get +6y & -6y
+
Could multiply A by 5 & B
by 3 to get 15x in each
Answer
x = 5, y = 4
Substitution Method
Given the following
equations :
y = x + 3 (i)
y = 2x
(ii)
Replace the y in
equation (i) with 2x
from equation (ii)
2x = x + 3
2x – x = 3
x=3
Sub. x = 3 into
either of the two
original equations to
find the value of y
y = x + 3 (i)
y=3+3
y=6
The answer is
(3, 6)
Substitution Method
A tool hire firm offers two
ways in which a tool may
be hired:
•Plan A - $20 a day
•Plan B - A payment of
$40 then $10 a day
Find the number of days
whereby there is no
difference in the cost of
hiring the tool from Plan A
and Plan B.
Let :
y = $ in hiring tool
x = no. of days hiring tool
y = 20x -----(1)
y = 40 + 10x -----(2)
Sub.(1) into (2)
20x = 40 + 10x
10x = 40
x = 4, y = 80
Graphical Method
x+y=6
Let x = 0
y=6
Coordinates (0, 6)
Let y = 0
x=6
Coordinates (6, 0)
2x + y = 8
Let x = 0
y=8
Coordinates (0, 8)
Let y = 0
2x = 8
x=4
Coordinates (4, 0)
1. x + y = 6
2x + y = 8
Graphical Method
y=x+3
Let x = 0
y=3
Coordinates (0, 3)
Let y = 0
x = -3
Coordinates (-3, 0)
y = 2x
Let x = 0
y=0
Coordinates (0, 0)
Let y = 4
2x = 4
x=2
Coordinates (2, 0)
y=x+3
y = 2x